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SIMO communication with impulsive and dependent
interference - the Copula receiver
Emilie Soret, Laurent Clavier, Gareth W. Peters, Ido Nevat, François Septier
To cite this version:
Emilie Soret, Laurent Clavier, Gareth W. Peters, Ido Nevat, François Septier. SIMO communication
with impulsive and dependent interference - the Copula receiver. XXVIème Colloque GRETSI, Sep
2017, Juan-les-Pins, France. �hal-01678984�
SIMO communication with impulsive and dependent interference
-the Copula receiver.
´
Emilie SORET1, Laurent CLAVIER1,2, Gareth W. PETERS3, Ido NEVAT4, Franc¸ois SEPTIER5
1Univ. Lille, CNRS, Centrale Lille, ISEN, Univ. Valenciennes, UMR 8520 - IEMN, F-59000 Lille, France 2IMT Lille Douai, IMT, F-59000 Lille, France
3Department of Statistical Science, University College London; London, UK. 4Institute for Infocom Research (I2R), A*STAR, Singapore
5IMT Lille Douai, Univ. Lille, CNRS, UMR 9189 - CRIStAL, F-59000 Lille, France
emilie.soret@ircica.univ-lille1.fr, laurent.clavier@telecom-lille.fr
R´esum´e –Dans ce papier, nous proposons une m´ethode pour mod´eliser la d´ependance entre des bruits impulsifs. Nous utilisons la notion de copule ce qui nous permet de repr´esenter les d´ependances d’upper et de lower tail, ce qui n’est pas le cas des coefficients de corr´elation classique (qui de plus, ne sont pas adapt´es aux lois α-stables, souvent utilis´ees pour mod´eliser des bruits impulsifs). Afin d’ illustrer l’approche par les copules, nous consid´erons une configuration de communication simple avec une antenne de transmission et deux antennes de r´eception. Nous pouvons alors construire un r´ecepteur adapt´e. Nous d´eterminons analytiquement le rapport de vraisemblance qui se d´ecompose en deux parties : une d´ependant uniquement des marginales et une d´ependant de la copule. Nous pouvons ensuite illustrer l’impact de la structure de d´ependance sur les r´egions de d´ecision et les performances du syst`emes.
Abstract –In this paper, we propose solutions for modelling dependence in impulsive noises. We use the copula framework that allows to represent the upper and lower tail dependencies that can not be captured by classical correlation (which, besides, is not adapted to α-stable distributions often considered in modelling impulsive noise). To illustrate the copula approach we consider a simple communication link with a single transmit antenna and two receive antennas and an adapted receiver architecture. We can derive the likelihood ratio that exhibits two components: one from the marginals and one from the copulas. We can then illustrate the impact of the dependence structure on the decision regions.
1
Introduction
Impulsive interference are encountered in many situations, e.g. power line communications, ultra-wide band technology, or in dense networks. This significantly degrades the perfor-mance of the classical receivers [1]. In this paper, we consider a simple detection problem in a block fading scenario. Each data symbol is transmitted over wireless channels and K = 2 ver-sions of each symbol are received. We only consider the case K= 2 for clear analytical expressions and simple illustrations but this can be extended to higher dimensions. This transmis-sion structure can be motivated by many different practical wi-reless communication systems like a rake receiver [2], a Single-Input-Multiple-Output (SIMO) system [3], a cooperative com-munication scheme involving multiple relays [4] or in impulse radio Ultra Wide Band systems with symbol repetitions [5].
For a single transmitted symbol sn at time n, the received
signal Y∈ RKis Y= s
nhn+ Ik+ Nn, where hn ∈ RK is
the block fading channel coefficients, In ∈ RKis the impulsive
interference and Nn i.i.d.
∼ N (0, σ2) is the thermal noise.
In this paper we make the assumption that the channel state
is perfectly known and that interference is dominating. Besides we assume independence between different time instant n so that we will drop this index for clarity. The studied case can then be summarized by Y= S + I, where S is a vector contai-ning the repeated sample s and I the interference vector.
Many papers have considered the case where I is compo-sed of independent and identically distributed samples. Depen-ding on the impulsive interference distribution assumption, it is more or less complicated to derive the optimal receiver and sometimes suboptimal approaches are considered [6].
In this paper we consider an α-stable interference distribu-tion but we do not consider any longer that the components of I are independent. We take the example of a SIMO link : if a strong interference is received on one antenna, the probability of receiving a strong interference sample on another antenna is not negligible. This upper tail dependence can not be captu-red by traditional correlation function that, anyway, can not be used for α-stable random vectors. We propose to use the copula framework to model the dependence structure. It allows to se-parately model the marginal distributions and the dependence structure.
2
Copulæ
Copulæ are a very useful way to model structures of depen-dence between random variables [7]. The fundamental result justifying this usefulness is the Sklar’s Theorem : it ensures that under the condition that the cumulative distributions of the random variables are continuous, there exists a unique copula C such that∀(x1, . . . , xd), we have
H(x1, . . . , xd) = C (F1(x1), . . . , Fd(xd))) . (1)
where H is the joint distribution of(X1, . . . , Xd). Hence, a
co-pula is a function C : [0, 1]d 7→ [0, 1] which couples the
mar-ginals Fi between themselves. The name copula comes from
this last remark. In Fig. 1 we represent the interference samples when I has independent components. The representation is done directly on the sample or after a transformation through the re-partition function of the marginals (Fi(.)) to have the
represen-tation of the copula.
−20 −10 0 10 20 −20 −10 0 10 20 y 1 y2 Received samples 0 0.5 1 0 0.2 0.4 0.6 0.8 1 F i(y1) Fi (y2 )
Received samples in the Copula space
FIGURE1 – In the left plot samples are independent and they form a cross. This can be explained saying that large values are rare and the occurrence of two large values on the same vector is very unlikely. In the right plot (X and Y axis are Fi(y1) and
Fi(y2)), the points are uniformly distributed which signifies the
independent structure.
2.1
Archimedean copulæ
In the following, we consider a particular class of bivariate Archimedean copulæ. The interest of this class is, first of all, the easiness with which they can be constructed. The multi-variate Archimedean copulæ have the following form : for all (u1,· · · , ud) ∈ [0, 1]d,
C(u1,· · · , ud) = φ−1(φ(u1) + · · · + φ(ud)) . (2)
The function φ is called the generator of the copula and is a continuous and convex function such that φ(1) = 0. It appears that all Archimedean copula is symmetric in its variables.
We will focus on two families of Archimedean copulæ, both indexed by a single parameter. The Clayton and the Gumbel families of copulæ model asymmetric dependence in tails. Definition 2.1. For all θ >0,The Clayton copula of parameter θ is defined on[0, 1]dby C(u1,· · · , ud) = u−11 /θ+ · · · + u −1θ d − (d − 1) −θ .
In particular, it is obtained when φ−1is the Laplace transform
of a Gamma distribution.
In Fig. 2 we have a similar representation as in Fig. 1 but in-troducing the dependence structure of the Clayton copula. The
0 0 y1 y2 Received samples 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Fi(y1) Fi (y2 )
Received samples in the copula plane
FIGURE2 – Interference samples for Cauchy marginals and Clayton copula.
cross in the left plot tends to disappear and points, especially in the bottom left quadrant, are differently positioned. This re-sults from the non zero asymmetric tail dependence introduced by the Clayton copula.
Definition 2.2. For all θ≥ 1,The Gumbel copula of parameter θ is defined on[0, 1]dby C(u1,· · · , ud) = exp −( d X i=1 (− log ui))1/θ ! .
In particular, it is obtained when φ−1is the Laplace transform
of a α-stable distribution.
In Fig. 3 we represent the dependence structure of the Gum-bel copula on the received samples and after the transform through the marginals. −10 −5 0 5 10 −10 −5 0 5 10 y1 y2 Received samples 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Fi(y1) Fi y2 )
Received samples in the Copula space
FIGURE3 – Received samples for Cauchy marginals and Gum-bel copula.
3
LLR for dependent variables
In the two-dimensional case and with a binary input, our sys-tem model in section 1can be written
(
y1= s + i1
y2= s + i2
, (3)
where s∈ {−1, 1}. Two repetitions y1and y2of this
transmit-ted bit are obtained and I = (i1, i2) is a bivariate interference
vector. The two coordinates i1and i2are not independent. The
Log Likelihoood Ratio (LLR) for each Y∈ R2
is given by Λ(y1, y2) = log P(y
1= s + i1, y2= s + i2| s = 1)
P(y1= s + i1, y2= s + i2| s = −1)
Let f be the joint density of the couple(i1, i2), (4) becomes Λ(y1, y2) = log f(y1− 1, y2− 1) f(y1+ 1, y2+ 1) . (5)
3.1
Independent interferences
In the left plot in Fig. 4, we illustrate the two decision re-gions in the case when interference is independent on the two dimensions and Cauchy distributed with location x0 = 0 and
scale δ = 1. The X and Y axis are the values of the com-ponents of the received vector Y. We consider two possible transmitted symbols{−1, 1} meaning that the transmitted vec-tor corresponds to the points(1, 1) and (−1, −1). The white region corresponds to the decision1, meaning that Λ ≥ 0 and the black one to−1, i.e., Λ < 0
A Gaussian noise would correspond to a linear boundary, corresponding to an Euclidean distance. Impulsiveness signi-ficantly modifies those boundaries and necessitate non linear operation to implement an optimal receiver.
3.2
Dependent interferences
If we now consider that i1and i2are dependent and that we
can express this dependence through an Archimedean copula, the LLR will become
Λ(x, y) = logfi(x − 1)fi(y − 1)c(Fi(x − 1), Fi(y − 1)) fi(x + 1)fi(y + 1)c(Fi(x + 1), Fi(y + 1))
= Λ⊥(x, y) + Λc(x, y), (6)
where c is the density of the copula and is defined by c(u, v) = ∂
2
C
∂u∂v(u, v); (7)
fiand Fiare respectively the probability density function and
the cumulative distribution of the interference.Λ⊥ represents
the independent part of the LLR. The second term Λc(x, y) = log
c(Fi(x − 1), Fi(y − 1))
c(Fi(x + 1), Fi(y + 1))
(8) is the part of the LLR depending on the copula and represents the dependence structure. It can however be tricky to derive because it also depends on the marginals.
In the case of the Clayton copula, the consequence on the decision region is shown in Fig. 4. We clearly see that the lower tail dependence significantly modifies the decision regions.
In the case of the Gumbel copula, the consequence on the decision region is shown in Fig. 5. We again clearly see that the lower tail dependence significantly modifies the decision regions.
4
Application to SIMO transmissions
4.1
Receiver design
The optimal receiver in terms of minimizing the Bit Error Rate (BER) is the Maximum Likelihood (ML) detectorsˆ =
Independent y1 y2 −4 −2 0 2 4 −4 −2 0 2 4 Dependent (Clayton, θ=1) y1 y2 −4 −2 0 2 4 −4 −2 0 2 4 Differences y1 y2 −4 −2 0 2 4 −4 −2 0 2 4
FIGURE4 – Decision region for independent Cauchy, Cauchy marginals and Clayton copula and the difference between both (in white the areas where the dependence structure modifies the optimal decision). Independent y1 y2 −4 −2 0 2 4 −4 −2 0 2 4 Dependent (Gumbel, θ=3) y1 y2 −4 −2 0 2 4 −4 −2 0 2 4 Differences y1 y2 −4 −2 0 2 4 −4 −2 0 2 4
FIGURE5 – Decision region for independent Cauchy, Cauchy marginals and Gumbel copula and the difference between both (in white the areas where the dependence structure modifies the optimal decision).
arg maxs∈ΩP(Y|s), where Ω is the set of possible
transmit-ted bits. In the binary case,Ω = {−1, 1} and the problem is reduced to obtaining the sign of the LLR defined in (6).
bs = sign (Λ(x, y)) = sign (log Λ⊥(x, y) + Λc(x, y)) , (9)
Fig. 6 compares the performance of the linear Gaussian re-ceiver, a Cauchy receiver assuming independent Cauchy inter-ference and a copula receiver that knows both the marginal and the dependence structures. In that case the dependence is cap-tured by a Clayton copula. Obviously when the parameter gets close to zero, the dependence is low, the Cauchy receiver out-performs the Gaussian receiver and there is no need to intro-duce the dependence structure. However, when the dependence increases (θ gets larger), the performance of the Cauchy recei-ver quickly degrades when the copula receirecei-ver is able to main-tain a better performance level.
1 2 3 −2.6 −2.4 −2.2 −2 −1.8
Clayton copula and Cauchy marginals S
1(0,0.05,0) θ log 10 (BER) Gaussian receiver Cauchy receiver Copula receiver
FIGURE6 – BER for Cauchy noise and Clayton copula as a function of the Clayton parameter.
Fig. 7 shows similar results with the Gumbel copula. We finally apply our Copula receiver to the SIMO case. In-terferers are uniformly distributed in a square around the
recei-1 3 5 −2.7
−2.3 −1.9
Gumbel copula and Cauchy marginals S1(0,0.05,0)
θ log 10 (BER) Gaussian receiver Cauchy receiver Copula receiver
FIGURE7 – BER for Cauchy noise and Gumbel copula as a function of the Gumbel parameter.
ver. The square is 10 by 10 and the mean number of interfe-rers is 50. We consider normalized distances so the unity is not significant. The channels are Rice channels with a main path strength randomly chosen. The channel attenuation coefficient is 3. For the copula receiver, we chose a mixture of the Gumbel and Clayton copulas to ensure a symmetry in the upper and lo-wer tail dependence. To have the symmetry, the parameters for the two copula are linked so that only one parameter has to be chosen to define the dependence structure. To observe the im-pact of including the dependence in the receiver, we vary this parameter. Fig. 8 shows the benefit of including the dependence structure to design the receiver.
0 1 2 3 4
−1.6 −1.55 −1.5 −1.45
Clayton copula and Cauchy marginals
θ log 10 (BER) Gaussian receiver Cauchy receiver Copula receiver
FIGURE8 – BER for SIMO model for mixture of Clayton and Gumbel copulæ.
The gain in performance is limited but this is easily explai-ned by the small dimension considered (only 2). Besides, we chose a Cauchy distribution for the marginals which is not the optimal choice. It has however proved to be close to the optimal in several situations [5]. It is clear that taken the dependence structure into account allow a further gain compared to the independent receiver, which already gives better performance than the Gaussian receiver.
5
Conclusion
We proposed in this paper a way to model dependency in impulsive interference. Usual tools (based on correlation) do not allow to well capture the dependence structure of such an impulsive interference, especially when the α-stable model is used.
In the case of Cauchy marginals and copulæ from the Ar-chimedean family and with a binary input, we are able to de-rive analytical expressions of the decision rule based on the likelihood ratio. The results on the decision regions show that dependent interference has a significant impact on the optimal decision that we should make. Consequently, we compared re-ceivers that takes this dependency into account to rere-ceivers that do not. We show that the latter can rapidly degrade if a depen-dence structure is present when the former manage to main-tain good performances. We illustrate the possible benefit on a SIMO example. Many questions are still open, what is the best choice for the copula ? How can we estimate the interference parameters ? How can we implement such a receiver, especially if the dimension increases ?
The densification of networks and their heterogeneity make interference an important issue in wireless communication. The dependence structure is certainly a crucial point for an efficient implementation of such networks. Will copula play a role ?
6
Acknowledgement
This work is part of the research project PERSEPTEUR (fi-nanced by the French Agence Nationale de la Recherche ANR) and supported by IRCICA, USR CNRS 3380.
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